LMFDB - Embedded newform 4598.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4598,2,Mod(1,4598)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4598, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4598.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4598 = 2 \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4598.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$36.7152148494$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.39138$ of defining polynomial
Character	$\chi$	$=$	4598.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} -2.39138 q^{6} +1.71871 q^{7} -1.00000 q^{8} +2.71871 q^{9} +0.391382 q^{10} +2.39138 q^{12} -3.71871 q^{13} -1.71871 q^{14} -0.935945 q^{15} +1.00000 q^{16} -5.43742 q^{17} -2.71871 q^{18} +1.00000 q^{19} -0.391382 q^{20} +4.11009 q^{21} -3.43742 q^{23} -2.39138 q^{24} -4.84682 q^{25} +3.71871 q^{26} -0.672673 q^{27} +1.71871 q^{28} +3.82880 q^{29} +0.935945 q^{30} -3.06406 q^{31} -1.00000 q^{32} +5.43742 q^{34} -0.672673 q^{35} +2.71871 q^{36} -4.65465 q^{37} -1.00000 q^{38} -8.89286 q^{39} +0.391382 q^{40} -1.06406 q^{41} -4.11009 q^{42} +3.73673 q^{43} -1.06406 q^{45} +3.43742 q^{46} -8.22018 q^{47} +2.39138 q^{48} -4.04604 q^{49} +4.84682 q^{50} -13.0029 q^{51} -3.71871 q^{52} -1.21724 q^{53} +0.672673 q^{54} -1.71871 q^{56} +2.39138 q^{57} -3.82880 q^{58} +12.4404 q^{59} -0.935945 q^{60} -2.00000 q^{61} +3.06406 q^{62} +4.67267 q^{63} +1.00000 q^{64} +1.45544 q^{65} -1.71871 q^{67} -5.43742 q^{68} -8.22018 q^{69} +0.672673 q^{70} +1.04604 q^{71} -2.71871 q^{72} -13.6576 q^{73} +4.65465 q^{74} -11.5906 q^{75} +1.00000 q^{76} +8.89286 q^{78} -13.0029 q^{79} -0.391382 q^{80} -9.76475 q^{81} +1.06406 q^{82} -8.51949 q^{83} +4.11009 q^{84} +2.12811 q^{85} -3.73673 q^{86} +9.15613 q^{87} +17.6936 q^{89} +1.06406 q^{90} -6.39138 q^{91} -3.43742 q^{92} -7.32733 q^{93} +8.22018 q^{94} -0.391382 q^{95} -2.39138 q^{96} -4.65465 q^{97} +4.04604 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} - q^{6} - q^{7} - 3 q^{8} + 2 q^{9} - 5 q^{10} + q^{12} - 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} - 4 q^{17} - 2 q^{18} + 3 q^{19} + 5 q^{20} + 2 q^{23} - q^{24}+ \cdots + 6 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 - q^6 - q^7 - 3 * q^8 + 2 * q^9 - 5 * q^10 + q^12 - 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 - 4 * q^17 - 2 * q^18 + 3 * q^19 + 5 * q^20 + 2 * q^23 - q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 - q^28 - 7 * q^29 + 9 * q^30 - 3 * q^31 - 3 * q^32 + 4 * q^34 - 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 - 2 * q^39 - 5 * q^40 + 3 * q^41 + 5 * q^43 + 3 * q^45 - 2 * q^46 + q^48 - 6 * q^49 - 4 * q^50 - 2 * q^51 - 5 * q^52 - 16 * q^53 + 2 * q^54 + q^56 + q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 - 6 * q^61 + 3 * q^62 + 14 * q^63 + 3 * q^64 - 8 * q^65 + q^67 - 4 * q^68 + 2 * q^70 - 3 * q^71 - 2 * q^72 - 4 * q^73 + 14 * q^74 - 41 * q^75 + 3 * q^76 + 2 * q^78 - 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 - 7 * q^83 - 6 * q^85 - 5 * q^86 + 9 * q^87 + 16 * q^89 - 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 + 5 * q^95 - q^96 - 14 * q^97 + 6 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.39138	1.38067	0.690333	−	0.723492i	$-0.257464\pi$
$3$	2.39138	1.38067	0.690333	+	0.723492i	$0.257464\pi$
$4$	1.00000	0.500000
$5$	−0.391382	−0.175032	−0.0875158	−	0.996163i	$-0.527893\pi$
$5$	−0.391382	−0.175032	−0.0875158	+	0.996163i	$0.527893\pi$
$6$	−2.39138	−0.976278
$7$	1.71871	0.649611	0.324806	−	0.945781i	$-0.394701\pi$
$7$	1.71871	0.649611	0.324806	+	0.945781i	$0.394701\pi$
$8$	−1.00000	−0.353553
$9$	2.71871	0.906237
$10$	0.391382	0.123766
$11$	0	0
$11$	0	0
$12$	2.39138	0.690333
$13$	−3.71871	−1.03138	−0.515692	−	0.856774i	$-0.672465\pi$
$13$	−3.71871	−1.03138	−0.515692	+	0.856774i	$0.672465\pi$
$14$	−1.71871	−0.459344
$15$	−0.935945	−0.241660
$16$	1.00000	0.250000
$17$	−5.43742	−1.31877	−0.659384	−	0.751806i	$-0.729183\pi$
$17$	−5.43742	−1.31877	−0.659384	+	0.751806i	$0.729183\pi$
$18$	−2.71871	−0.640806
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.391382	−0.0875158
$21$	4.11009	0.896896
$22$	0	0
$23$	−3.43742	−0.716751	−0.358376	−	0.933577i	$-0.616669\pi$
$23$	−3.43742	−0.716751	−0.358376	+	0.933577i	$0.616669\pi$
$24$	−2.39138	−0.488139
$25$	−4.84682	−0.969364
$26$	3.71871	0.729299
$27$	−0.672673	−0.129456
$28$	1.71871	0.324806
$29$	3.82880	0.710991	0.355495	−	0.934678i	$-0.384312\pi$
$29$	3.82880	0.710991	0.355495	+	0.934678i	$0.384312\pi$
$30$	0.935945	0.170879
$31$	−3.06406	−0.550321	−0.275160	−	0.961398i	$-0.588731\pi$
$31$	−3.06406	−0.550321	−0.275160	+	0.961398i	$0.588731\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.43742	0.932510
$35$	−0.672673	−0.113702
$36$	2.71871	0.453118
$37$	−4.65465	−0.765221	−0.382610	−	0.923910i	$-0.624975\pi$
$37$	−4.65465	−0.765221	−0.382610	+	0.923910i	$0.624975\pi$
$38$	−1.00000	−0.162221
$39$	−8.89286	−1.42400
$40$	0.391382	0.0618830
$41$	−1.06406	−0.166177	−0.0830887	−	0.996542i	$-0.526478\pi$
$41$	−1.06406	−0.166177	−0.0830887	+	0.996542i	$0.526478\pi$
$42$	−4.11009	−0.634201
$43$	3.73673	0.569846	0.284923	−	0.958550i	$-0.408032\pi$
$43$	3.73673	0.569846	0.284923	+	0.958550i	$0.408032\pi$
$44$	0	0
$45$	−1.06406	−0.158620
$46$	3.43742	0.506820
$47$	−8.22018	−1.19904	−0.599519	−	0.800361i	$-0.704641\pi$
$47$	−8.22018	−1.19904	−0.599519	+	0.800361i	$0.704641\pi$
$48$	2.39138	0.345166
$49$	−4.04604	−0.578005
$50$	4.84682	0.685444
$51$	−13.0029	−1.82078
$52$	−3.71871	−0.515692
$53$	−1.21724	−0.167200	−0.0836001	−	0.996499i	$-0.526642\pi$
$53$	−1.21724	−0.167200	−0.0836001	+	0.996499i	$0.526642\pi$
$54$	0.672673	0.0915392
$55$	0	0
$56$	−1.71871	−0.229672
$57$	2.39138	0.316746
$58$	−3.82880	−0.502746
$59$	12.4404	1.61960	0.809799	−	0.586707i	$-0.199576\pi$
$59$	12.4404	1.61960	0.809799	+	0.586707i	$0.199576\pi$
$60$	−0.935945	−0.120830
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	3.06406	0.389135
$63$	4.67267	0.588701
$64$	1.00000	0.125000
$65$	1.45544	0.180525
$66$	0	0
$67$	−1.71871	−0.209974	−0.104987	−	0.994474i	$-0.533480\pi$
$67$	−1.71871	−0.209974	−0.104987	+	0.994474i	$0.533480\pi$
$68$	−5.43742	−0.659384
$69$	−8.22018	−0.989594
$70$	0.672673	0.0803998
$71$	1.04604	0.124142	0.0620709	−	0.998072i	$-0.480230\pi$
$71$	1.04604	0.124142	0.0620709	+	0.998072i	$0.480230\pi$
$72$	−2.71871	−0.320403
$73$	−13.6576	−1.59850	−0.799251	−	0.600998i	$-0.794770\pi$
$73$	−13.6576	−1.59850	−0.799251	+	0.600998i	$0.794770\pi$
$74$	4.65465	0.541093
$75$	−11.5906	−1.33837
$76$	1.00000	0.114708
$77$	0	0
$78$	8.89286	1.00692
$79$	−13.0029	−1.46295	−0.731473	−	0.681870i	$-0.761167\pi$
$79$	−13.0029	−1.46295	−0.731473	+	0.681870i	$0.761167\pi$
$80$	−0.391382	−0.0437579
$81$	−9.76475	−1.08497
$82$	1.06406	0.117505
$83$	−8.51949	−0.935136	−0.467568	−	0.883957i	$-0.654870\pi$
$83$	−8.51949	−0.935136	−0.467568	+	0.883957i	$0.654870\pi$
$84$	4.11009	0.448448
$85$	2.12811	0.230826
$86$	−3.73673	−0.402942
$87$	9.15613	0.981640
$88$	0	0
$89$	17.6936	1.87552	0.937761	−	0.347281i	$-0.112895\pi$
$89$	17.6936	1.87552	0.937761	+	0.347281i	$0.112895\pi$
$90$	1.06406	0.112161
$91$	−6.39138	−0.669999
$92$	−3.43742	−0.358376
$93$	−7.32733	−0.759808
$94$	8.22018	0.847847
$95$	−0.391382	−0.0401550
$96$	−2.39138	−0.244069
$97$	−4.65465	−0.472609	−0.236304	−	0.971679i	$-0.575936\pi$
$97$	−4.65465	−0.472609	−0.236304	+	0.971679i	$0.575936\pi$
$98$	4.04604	0.408711
$99$	0	0
$100$	−4.84682	−0.484682
$101$	9.65760	0.960967	0.480484	−	0.877004i	$-0.340461\pi$
$101$	9.65760	0.960967	0.480484	+	0.877004i	$0.340461\pi$
$102$	13.0029	1.28748
$103$	−6.95396	−0.685194	−0.342597	−	0.939482i	$-0.611307\pi$
$103$	−6.95396	−0.685194	−0.342597	+	0.939482i	$0.611307\pi$
$104$	3.71871	0.364649
$105$	−1.60862	−0.156985
$106$	1.21724	0.118228
$107$	−2.65465	−0.256635	−0.128318	−	0.991733i	$-0.540958\pi$
$107$	−2.65465	−0.256635	−0.128318	+	0.991733i	$0.540958\pi$
$108$	−0.672673	−0.0647280
$109$	−6.00000	−0.574696	−0.287348	−	0.957826i	$-0.592774\pi$
$109$	−6.00000	−0.574696	−0.287348	+	0.957826i	$0.592774\pi$
$110$	0	0
$111$	−11.1311	−1.05651
$112$	1.71871	0.162403
$113$	7.52949	0.708315	0.354158	−	0.935186i	$-0.384768\pi$
$113$	7.52949	0.708315	0.354158	+	0.935186i	$0.384768\pi$
$114$	−2.39138	−0.223973
$115$	1.34535	0.125454
$116$	3.82880	0.355495
$117$	−10.1101	−0.934678
$118$	−12.4404	−1.14523
$119$	−9.34535	−0.856686
$120$	0.935945	0.0854397
$121$	0	0
$122$	2.00000	0.181071
$123$	−2.54456	−0.229435
$124$	−3.06406	−0.275160
$125$	3.85387	0.344701
$126$	−4.67267	−0.416275
$127$	5.00295	0.443940	0.221970	−	0.975054i	$-0.428751\pi$
$127$	5.00295	0.443940	0.221970	+	0.975054i	$0.428751\pi$
$128$	−1.00000	−0.0883883
$129$	8.93594	0.786766
$130$	−1.45544	−0.127650
$131$	6.50147	0.568036	0.284018	−	0.958819i	$-0.408332\pi$
$131$	6.50147	0.568036	0.284018	+	0.958819i	$0.408332\pi$
$132$	0	0
$133$	1.71871	0.149031
$134$	1.71871	0.148474
$135$	0.263272	0.0226589
$136$	5.43742	0.466255
$137$	15.7187	1.34294	0.671470	−	0.741032i	$-0.265663\pi$
$137$	15.7187	1.34294	0.671470	+	0.741032i	$0.265663\pi$
$138$	8.22018	0.699749
$139$	11.1381	0.944722	0.472361	−	0.881405i	$-0.343402\pi$
$139$	11.1381	0.944722	0.472361	+	0.881405i	$0.343402\pi$
$140$	−0.672673	−0.0568512
$141$	−19.6576	−1.65547
$142$	−1.04604	−0.0877815
$143$	0	0
$144$	2.71871	0.226559
$145$	−1.49853	−0.124446
$146$	13.6576	1.13031
$147$	−9.67562	−0.798032
$148$	−4.65465	−0.382610
$149$	−11.3453	−0.929447	−0.464723	−	0.885456i	$-0.653846\pi$
$149$	−11.3453	−0.929447	−0.464723	+	0.885456i	$0.653846\pi$
$150$	11.5906	0.946368
$151$	14.6547	1.19258	0.596289	−	0.802770i	$-0.296641\pi$
$151$	14.6547	1.19258	0.596289	+	0.802770i	$0.296641\pi$
$152$	−1.00000	−0.0811107
$153$	−14.7828	−1.19512
$154$	0	0
$155$	1.19922	0.0963234
$156$	−8.89286	−0.711998
$157$	−10.4835	−0.836671	−0.418335	−	0.908293i	$-0.637386\pi$
$157$	−10.4835	−0.836671	−0.418335	+	0.908293i	$0.637386\pi$
$158$	13.0029	1.03446
$159$	−2.91087	−0.230847
$160$	0.391382	0.0309415
$161$	−5.90793	−0.465610
$162$	9.76475	0.767191
$163$	−3.47346	−0.272062	−0.136031	−	0.990705i	$-0.543435\pi$
$163$	−3.47346	−0.272062	−0.136031	+	0.990705i	$0.543435\pi$
$164$	−1.06406	−0.0830887
$165$	0	0
$166$	8.51949	0.661241
$167$	−23.0950	−1.78715	−0.893573	−	0.448917i	$-0.851810\pi$
$167$	−23.0950	−1.78715	−0.893573	+	0.448917i	$0.851810\pi$
$168$	−4.11009	−0.317100
$169$	0.828802	0.0637540
$170$	−2.12811	−0.163219
$171$	2.71871	0.207905
$172$	3.73673	0.284923
$173$	18.1771	1.38198	0.690990	−	0.722865i	$-0.257175\pi$
$173$	18.1771	1.38198	0.690990	+	0.722865i	$0.257175\pi$
$174$	−9.15613	−0.694124
$175$	−8.33028	−0.629710
$176$	0	0
$177$	29.7497	2.23612
$178$	−17.6936	−1.32619
$179$	11.2842	0.843424	0.421712	−	0.906730i	$-0.361429\pi$
$179$	11.2842	0.843424	0.421712	+	0.906730i	$0.361429\pi$
$180$	−1.06406	−0.0793100
$181$	21.8778	1.62616	0.813082	−	0.582150i	$-0.197788\pi$
$181$	21.8778	1.62616	0.813082	+	0.582150i	$0.197788\pi$
$182$	6.39138	0.473761
$183$	−4.78276	−0.353552
$184$	3.43742	0.253410
$185$	1.82175	0.133938
$186$	7.32733	0.537266
$187$	0	0
$188$	−8.22018	−0.599519
$189$	−1.15613	−0.0840960
$190$	0.391382	0.0283939
$191$	−7.09502	−0.513378	−0.256689	−	0.966494i	$-0.582632\pi$
$191$	−7.09502	−0.513378	−0.256689	+	0.966494i	$0.582632\pi$
$192$	2.39138	0.172583
$193$	−12.6476	−0.910394	−0.455197	−	0.890391i	$-0.650431\pi$
$193$	−12.6476	−0.910394	−0.455197	+	0.890391i	$0.650431\pi$
$194$	4.65465	0.334185
$195$	3.48051	0.249244
$196$	−4.04604	−0.289003
$197$	−8.43447	−0.600931	−0.300466	−	0.953793i	$-0.597142\pi$
$197$	−8.43447	−0.600931	−0.300466	+	0.953793i	$0.597142\pi$
$198$	0	0
$199$	−2.09207	−0.148303	−0.0741516	−	0.997247i	$-0.523625\pi$
$199$	−2.09207	−0.148303	−0.0741516	+	0.997247i	$0.523625\pi$
$200$	4.84682	0.342722
$201$	−4.11009	−0.289904
$202$	−9.65760	−0.679507
$203$	6.58060	0.461867
$204$	−13.0029	−0.910389
$205$	0.416452	0.0290863
$206$	6.95396	0.484506
$207$	−9.34535	−0.649546
$208$	−3.71871	−0.257846
$209$	0	0
$210$	1.60862	0.111005
$211$	6.09207	0.419396	0.209698	−	0.977766i	$-0.432752\pi$
$211$	6.09207	0.419396	0.209698	+	0.977766i	$0.432752\pi$
$212$	−1.21724	−0.0836001
$213$	2.50147	0.171398
$214$	2.65465	0.181468
$215$	−1.46249	−0.0997409
$216$	0.672673	0.0457696
$217$	−5.26622	−0.357494
$218$	6.00000	0.406371
$219$	−32.6606	−2.20700
$220$	0	0
$221$	20.2202	1.36016
$222$	11.1311	0.747068
$223$	−18.8748	−1.26395	−0.631976	−	0.774988i	$-0.717756\pi$
$223$	−18.8748	−1.26395	−0.631976	+	0.774988i	$0.717756\pi$
$224$	−1.71871	−0.114836
$225$	−13.1771	−0.878473
$226$	−7.52949	−0.500854
$227$	−20.9669	−1.39162	−0.695811	−	0.718225i	$-0.744955\pi$
$227$	−20.9669	−1.39162	−0.695811	+	0.718225i	$0.744955\pi$
$228$	2.39138	0.158373
$229$	13.8108	0.912642	0.456321	−	0.889815i	$-0.349167\pi$
$229$	13.8108	0.912642	0.456321	+	0.889815i	$0.349167\pi$
$230$	−1.34535	−0.0887094
$231$	0	0
$232$	−3.82880	−0.251373
$233$	−4.91087	−0.321722	−0.160861	−	0.986977i	$-0.551427\pi$
$233$	−4.91087	−0.321722	−0.160861	+	0.986977i	$0.551427\pi$
$234$	10.1101	0.660917
$235$	3.21724	0.209869
$236$	12.4404	0.809799
$237$	−31.0950	−2.01984
$238$	9.34535	0.605769
$239$	14.0490	0.908753	0.454377	−	0.890810i	$-0.349862\pi$
$239$	14.0490	0.908753	0.454377	+	0.890810i	$0.349862\pi$
$240$	−0.935945	−0.0604150
$241$	−22.7397	−1.46479	−0.732396	−	0.680879i	$-0.761598\pi$
$241$	−22.7397	−1.46479	−0.732396	+	0.680879i	$0.761598\pi$
$242$	0	0
$243$	−21.3332	−1.36853
$244$	−2.00000	−0.128037
$245$	1.58355	0.101169
$246$	2.54456	0.162235
$247$	−3.71871	−0.236616
$248$	3.06406	0.194568
$249$	−20.3734	−1.29111
$250$	−3.85387	−0.243740
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	4.67267	0.294351
$253$	0	0
$254$	−5.00295	−0.313913
$255$	5.08913	0.318693
$256$	1.00000	0.0625000
$257$	−19.4433	−1.21284	−0.606420	−	0.795144i	$-0.707395\pi$
$257$	−19.4433	−1.21284	−0.606420	+	0.795144i	$0.707395\pi$
$258$	−8.93594	−0.556328
$259$	−8.00000	−0.497096
$260$	1.45544	0.0902624
$261$	10.4094	0.644326
$262$	−6.50147	−0.401662
$263$	−14.6476	−0.903210	−0.451605	−	0.892218i	$-0.649148\pi$
$263$	−14.6476	−0.903210	−0.451605	+	0.892218i	$0.649148\pi$
$264$	0	0
$265$	0.476404	0.0292653
$266$	−1.71871	−0.105381
$267$	42.3123	2.58947
$268$	−1.71871	−0.104987
$269$	8.34829	0.509004	0.254502	−	0.967072i	$-0.418088\pi$
$269$	8.34829	0.509004	0.254502	+	0.967072i	$0.418088\pi$
$270$	−0.263272	−0.0160222
$271$	10.5375	0.640108	0.320054	−	0.947399i	$-0.396299\pi$
$271$	10.5375	0.640108	0.320054	+	0.947399i	$0.396299\pi$
$272$	−5.43742	−0.329692
$273$	−15.2842	−0.925044
$274$	−15.7187	−0.949602
$275$	0	0
$276$	−8.22018	−0.494797
$277$	22.8807	1.37477	0.687385	−	0.726293i	$-0.258758\pi$
$277$	22.8807	1.37477	0.687385	+	0.726293i	$0.258758\pi$
$278$	−11.1381	−0.668020
$279$	−8.33028	−0.498721
$280$	0.672673	0.0401999
$281$	4.39138	0.261968	0.130984	−	0.991384i	$-0.458186\pi$
$281$	4.39138	0.261968	0.130984	+	0.991384i	$0.458186\pi$
$282$	19.6576	1.17059
$283$	24.0670	1.43063	0.715317	−	0.698800i	$-0.246282\pi$
$283$	24.0670	1.43063	0.715317	+	0.698800i	$0.246282\pi$
$284$	1.04604	0.0620709
$285$	−0.935945	−0.0554406
$286$	0	0
$287$	−1.82880	−0.107951
$288$	−2.71871	−0.160202
$289$	12.5655	0.739149
$290$	1.49853	0.0879965
$291$	−11.1311	−0.652514
$292$	−13.6576	−0.799251
$293$	27.2901	1.59431	0.797153	−	0.603777i	$-0.206338\pi$
$293$	27.2901	1.59431	0.797153	+	0.603777i	$0.206338\pi$
$294$	9.67562	0.564294
$295$	−4.86894	−0.283481
$296$	4.65465	0.270546
$297$	0	0
$298$	11.3453	0.657218
$299$	12.7828	0.739246
$300$	−11.5906	−0.669184
$301$	6.42235	0.370178
$302$	−14.6547	−0.843281
$303$	23.0950	1.32677
$304$	1.00000	0.0573539
$305$	0.782765	0.0448210
$306$	14.7828	0.845074
$307$	9.22313	0.526392	0.263196	−	0.964742i	$-0.415223\pi$
$307$	9.22313	0.526392	0.263196	+	0.964742i	$0.415223\pi$
$308$	0	0
$309$	−16.6296	−0.946024
$310$	−1.19922	−0.0681110
$311$	6.34829	0.359979	0.179989	−	0.983669i	$-0.442394\pi$
$311$	6.34829	0.359979	0.179989	+	0.983669i	$0.442394\pi$
$312$	8.89286	0.503459
$313$	13.3943	0.757092	0.378546	−	0.925582i	$-0.376424\pi$
$313$	13.3943	0.757092	0.378546	+	0.925582i	$0.376424\pi$
$314$	10.4835	0.591616
$315$	−1.82880	−0.103041
$316$	−13.0029	−0.731473
$317$	31.0029	1.74130	0.870650	−	0.491904i	$-0.163699\pi$
$317$	31.0029	1.74130	0.870650	+	0.491904i	$0.163699\pi$
$318$	2.91087	0.163234
$319$	0	0
$320$	−0.391382	−0.0218789
$321$	−6.34829	−0.354327
$322$	5.90793	0.329236
$323$	−5.43742	−0.302546
$324$	−9.76475	−0.542486
$325$	18.0239	0.999787
$326$	3.47346	0.192377
$327$	−14.3483	−0.793462
$328$	1.06406	0.0587526
$329$	−14.1281	−0.778908
$330$	0	0
$331$	−16.2993	−0.895891	−0.447946	−	0.894061i	$-0.647844\pi$
$331$	−16.2993	−0.895891	−0.447946	+	0.894061i	$0.647844\pi$
$332$	−8.51949	−0.467568
$333$	−12.6547	−0.693471
$334$	23.0950	1.26370
$335$	0.672673	0.0367520
$336$	4.11009	0.224224
$337$	3.19217	0.173888	0.0869442	−	0.996213i	$-0.472290\pi$
$337$	3.19217	0.173888	0.0869442	+	0.996213i	$0.472290\pi$
$338$	−0.828802	−0.0450809
$339$	18.0059	0.977946
$340$	2.12811	0.115413
$341$	0	0
$342$	−2.71871	−0.147011
$343$	−18.9849	−1.02509
$344$	−3.73673	−0.201471
$345$	3.21724	0.173210
$346$	−18.1771	−0.977207
$347$	−24.6966	−1.32578	−0.662891	−	0.748716i	$-0.730671\pi$
$347$	−24.6966	−1.32578	−0.662891	+	0.748716i	$0.730671\pi$
$348$	9.15613	0.490820
$349$	4.43447	0.237372	0.118686	−	0.992932i	$-0.462132\pi$
$349$	4.43447	0.237372	0.118686	+	0.992932i	$0.462132\pi$
$350$	8.33028	0.445272
$351$	2.50147	0.133519
$352$	0	0
$353$	25.3152	1.34739	0.673696	−	0.739008i	$-0.264706\pi$
$353$	25.3152	1.34739	0.673696	+	0.739008i	$0.264706\pi$
$354$	−29.7497	−1.58118
$355$	−0.409400	−0.0217287
$356$	17.6936	0.937761
$357$	−22.3483	−1.18280
$358$	−11.2842	−0.596391
$359$	13.9749	0.737569	0.368784	−	0.929515i	$-0.379774\pi$
$359$	13.9749	0.737569	0.368784	+	0.929515i	$0.379774\pi$
$360$	1.06406	0.0560806
$361$	1.00000	0.0526316
$362$	−21.8778	−1.14987
$363$	0	0
$364$	−6.39138	−0.334999
$365$	5.34535	0.279788
$366$	4.78276	0.249999
$367$	−25.3453	−1.32302	−0.661508	−	0.749938i	$-0.730083\pi$
$367$	−25.3453	−1.32302	−0.661508	+	0.749938i	$0.730083\pi$
$368$	−3.43742	−0.179188
$369$	−2.89286	−0.150596
$370$	−1.82175	−0.0947083
$371$	−2.09207	−0.108615
$372$	−7.32733	−0.379904
$373$	−8.95396	−0.463619	−0.231809	−	0.972761i	$-0.574465\pi$
$373$	−8.95396	−0.463619	−0.231809	+	0.972761i	$0.574465\pi$
$374$	0	0
$375$	9.21608	0.475916
$376$	8.22018	0.423924
$377$	−14.2382	−0.733305
$378$	1.15613	0.0594649
$379$	−25.8527	−1.32796	−0.663982	−	0.747748i	$-0.731135\pi$
$379$	−25.8527	−1.32796	−0.663982	+	0.747748i	$0.731135\pi$
$380$	−0.391382	−0.0200775
$381$	11.9640	0.612932
$382$	7.09502	0.363013
$383$	−20.6296	−1.05412	−0.527061	−	0.849827i	$-0.676706\pi$
$383$	−20.6296	−1.05412	−0.527061	+	0.849827i	$0.676706\pi$
$384$	−2.39138	−0.122035
$385$	0	0
$386$	12.6476	0.643746
$387$	10.1591	0.516415
$388$	−4.65465	−0.236304
$389$	15.6086	0.791388	0.395694	−	0.918382i	$-0.370504\pi$
$389$	15.6086	0.791388	0.395694	+	0.918382i	$0.370504\pi$
$390$	−3.48051	−0.176242
$391$	18.6907	0.945229
$392$	4.04604	0.204356
$393$	15.5475	0.784268
$394$	8.43447	0.424922
$395$	5.08913	0.256062
$396$	0	0
$397$	−34.2512	−1.71902	−0.859508	−	0.511122i	$-0.829230\pi$
$397$	−34.2512	−1.71902	−0.859508	+	0.511122i	$0.829230\pi$
$398$	2.09207	0.104866
$399$	4.11009	0.205762
$400$	−4.84682	−0.242341
$401$	4.09207	0.204348	0.102174	−	0.994767i	$-0.467420\pi$
$401$	4.09207	0.204348	0.102174	+	0.994767i	$0.467420\pi$
$402$	4.11009	0.204993
$403$	11.3943	0.567592
$404$	9.65760	0.480484
$405$	3.82175	0.189904
$406$	−6.58060	−0.326590
$407$	0	0
$408$	13.0029	0.643742
$409$	−2.48346	−0.122799	−0.0613995	−	0.998113i	$-0.519556\pi$
$409$	−2.48346	−0.122799	−0.0613995	+	0.998113i	$0.519556\pi$
$410$	−0.416452	−0.0205671
$411$	37.5894	1.85415
$412$	−6.95396	−0.342597
$413$	21.3814	1.05211
$414$	9.34535	0.459299
$415$	3.33438	0.163678
$416$	3.71871	0.182325
$417$	26.6355	1.30435
$418$	0	0
$419$	35.0950	1.71450	0.857252	−	0.514897i	$-0.172170\pi$
$419$	35.0950	1.71450	0.857252	+	0.514897i	$0.172170\pi$
$420$	−1.60862	−0.0784925
$421$	−12.3483	−0.601819	−0.300910	−	0.953653i	$-0.597290\pi$
$421$	−12.3483	−0.601819	−0.300910	+	0.953653i	$0.597290\pi$
$422$	−6.09207	−0.296558
$423$	−22.3483	−1.08661
$424$	1.21724	0.0591142
$425$	26.3542	1.27837
$426$	−2.50147	−0.121197
$427$	−3.43742	−0.166348
$428$	−2.65465	−0.128318
$429$	0	0
$430$	1.46249	0.0705275
$431$	6.94691	0.334621	0.167310	−	0.985904i	$-0.446492\pi$
$431$	6.94691	0.334621	0.167310	+	0.985904i	$0.446492\pi$
$432$	−0.672673	−0.0323640
$433$	−13.2533	−0.636912	−0.318456	−	0.947938i	$-0.603164\pi$
$433$	−13.2533	−0.636912	−0.318456	+	0.947938i	$0.603164\pi$
$434$	5.26622	0.252787
$435$	−3.58355	−0.171818
$436$	−6.00000	−0.287348
$437$	−3.43742	−0.164434
$438$	32.6606	1.56058
$439$	−15.3152	−0.730955	−0.365477	−	0.930820i	$-0.619094\pi$
$439$	−15.3152	−0.730955	−0.365477	+	0.930820i	$0.619094\pi$
$440$	0	0
$441$	−11.0000	−0.523810
$442$	−20.2202	−0.961776
$443$	−20.8807	−0.992074	−0.496037	−	0.868301i	$-0.665212\pi$
$443$	−20.8807	−0.992074	−0.496037	+	0.868301i	$0.665212\pi$
$444$	−11.1311	−0.528257
$445$	−6.92498	−0.328275
$446$	18.8748	0.893750
$447$	−27.1311	−1.28326
$448$	1.71871	0.0812014
$449$	−28.7887	−1.35862	−0.679310	−	0.733851i	$-0.737721\pi$
$449$	−28.7887	−1.35862	−0.679310	+	0.733851i	$0.737721\pi$
$450$	13.1771	0.621174
$451$	0	0
$452$	7.52949	0.354158
$453$	35.0449	1.64655
$454$	20.9669	0.984026
$455$	2.50147	0.117271
$456$	−2.39138	−0.111987
$457$	32.7526	1.53210	0.766052	−	0.642779i	$-0.222219\pi$
$457$	32.7526	1.53210	0.766052	+	0.642779i	$0.222219\pi$
$458$	−13.8108	−0.645336
$459$	3.65760	0.170722
$460$	1.34535	0.0627271
$461$	−20.2261	−0.942023	−0.471011	−	0.882127i	$-0.656111\pi$
$461$	−20.2261	−0.942023	−0.471011	+	0.882127i	$0.656111\pi$
$462$	0	0
$463$	−18.6907	−0.868630	−0.434315	−	0.900761i	$-0.643010\pi$
$463$	−18.6907	−0.868630	−0.434315	+	0.900761i	$0.643010\pi$
$464$	3.82880	0.177748
$465$	2.86779	0.132990
$466$	4.91087	0.227492
$467$	34.5685	1.59964	0.799819	−	0.600241i	$-0.204929\pi$
$467$	34.5685	1.59964	0.799819	+	0.600241i	$0.204929\pi$
$468$	−10.1101	−0.467339
$469$	−2.95396	−0.136401
$470$	−3.21724	−0.148400
$471$	−25.0700	−1.15516
$472$	−12.4404	−0.572614
$473$	0	0
$474$	31.0950	1.42824
$475$	−4.84682	−0.222387
$476$	−9.34535	−0.428343
$477$	−3.30931	−0.151523
$478$	−14.0490	−0.642586
$479$	17.7187	0.809589	0.404794	−	0.914408i	$-0.367343\pi$
$479$	17.7187	0.809589	0.404794	+	0.914408i	$0.367343\pi$
$480$	0.935945	0.0427198
$481$	17.3093	0.789237
$482$	22.7397	1.03576
$483$	−14.1281	−0.642851
$484$	0	0
$485$	1.82175	0.0827214
$486$	21.3332	0.967695
$487$	37.7426	1.71028	0.855141	−	0.518396i	$-0.173471\pi$
$487$	37.7426	1.71028	0.855141	+	0.518396i	$0.173471\pi$
$488$	2.00000	0.0905357
$489$	−8.30636	−0.375627
$490$	−1.58355	−0.0715374
$491$	−0.373364	−0.0168497	−0.00842485	−	0.999965i	$-0.502682\pi$
$491$	−0.373364	−0.0168497	−0.00842485	+	0.999965i	$0.502682\pi$
$492$	−2.54456	−0.114718
$493$	−20.8188	−0.937632
$494$	3.71871	0.167313
$495$	0	0
$496$	−3.06406	−0.137580
$497$	1.79783	0.0806439
$498$	20.3734	0.912952
$499$	−14.8388	−0.664276	−0.332138	−	0.943231i	$-0.607770\pi$
$499$	−14.8388	−0.664276	−0.332138	+	0.943231i	$0.607770\pi$
$500$	3.85387	0.172350
$501$	−55.2290	−2.46745
$502$	12.0000	0.535586
$503$	−33.2662	−1.48327	−0.741634	−	0.670805i	$-0.765949\pi$
$503$	−33.2662	−1.48327	−0.741634	+	0.670805i	$0.765949\pi$
$504$	−4.67267	−0.208137
$505$	−3.77982	−0.168200
$506$	0	0
$507$	1.98198	0.0880229
$508$	5.00295	0.221970
$509$	−8.47051	−0.375449	−0.187724	−	0.982222i	$-0.560111\pi$
$509$	−8.47051	−0.375449	−0.187724	+	0.982222i	$0.560111\pi$
$510$	−5.08913	−0.225350
$511$	−23.4735	−1.03840
$512$	−1.00000	−0.0441942
$513$	−0.672673	−0.0296992
$514$	19.4433	0.857608
$515$	2.72166	0.119931
$516$	8.93594	0.393383
$517$	0	0
$518$	8.00000	0.351500
$519$	43.4684	1.90805
$520$	−1.45544	−0.0638252
$521$	−37.3152	−1.63481	−0.817404	−	0.576064i	$-0.804588\pi$
$521$	−37.3152	−1.63481	−0.817404	+	0.576064i	$0.804588\pi$
$522$	−10.4094	−0.455607
$523$	31.6576	1.38429	0.692145	−	0.721758i	$-0.256666\pi$
$523$	31.6576	1.38429	0.692145	+	0.721758i	$0.256666\pi$
$524$	6.50147	0.284018
$525$	−19.9209	−0.869418
$526$	14.6476	0.638666
$527$	16.6606	0.725745
$528$	0	0
$529$	−11.1841	−0.486267
$530$	−0.476404	−0.0206937
$531$	33.8217	1.46774
$532$	1.71871	0.0745155
$533$	3.95691	0.171393
$534$	−42.3123	−1.83103
$535$	1.03899	0.0449192
$536$	1.71871	0.0742370
$537$	26.9849	1.16449
$538$	−8.34829	−0.359921
$539$	0	0
$540$	0.263272	0.0113294
$541$	−28.0921	−1.20777	−0.603886	−	0.797070i	$-0.706382\pi$
$541$	−28.0921	−1.20777	−0.603886	+	0.797070i	$0.706382\pi$
$542$	−10.5375	−0.452625
$543$	52.3182	2.24519
$544$	5.43742	0.233127
$545$	2.34829	0.100590
$546$	15.2842	0.654105
$547$	−24.1841	−1.03404	−0.517020	−	0.855973i	$-0.672959\pi$
$547$	−24.1841	−1.03404	−0.517020	+	0.855973i	$0.672959\pi$
$548$	15.7187	0.671470
$549$	−5.43742	−0.232063
$550$	0	0
$551$	3.82880	0.163112
$552$	8.22018	0.349874
$553$	−22.3483	−0.950346
$554$	−22.8807	−0.972109
$555$	4.35650	0.184923
$556$	11.1381	0.472361
$557$	−39.9699	−1.69358	−0.846789	−	0.531929i	$-0.821467\pi$
$557$	−39.9699	−1.69358	−0.846789	+	0.531929i	$0.821467\pi$
$558$	8.33028	0.352649
$559$	−13.8958	−0.587730
$560$	−0.672673	−0.0284256
$561$	0	0
$562$	−4.39138	−0.185239
$563$	−16.4764	−0.694398	−0.347199	−	0.937792i	$-0.612867\pi$
$563$	−16.4764	−0.694398	−0.347199	+	0.937792i	$0.612867\pi$
$564$	−19.6576	−0.827734
$565$	−2.94691	−0.123977
$566$	−24.0670	−1.01161
$567$	−16.7828	−0.704810
$568$	−1.04604	−0.0438907
$569$	24.9418	1.04562	0.522808	−	0.852450i	$-0.324884\pi$
$569$	24.9418	1.04562	0.522808	+	0.852450i	$0.324884\pi$
$570$	0.935945	0.0392024
$571$	8.50737	0.356022	0.178011	−	0.984028i	$-0.443034\pi$
$571$	8.50737	0.356022	0.178011	+	0.984028i	$0.443034\pi$
$572$	0	0
$573$	−16.9669	−0.708803
$574$	1.82880	0.0763327
$575$	16.6606	0.694793
$576$	2.71871	0.113280
$577$	31.8527	1.32605	0.663023	−	0.748599i	$-0.269273\pi$
$577$	31.8527	1.32605	0.663023	+	0.748599i	$0.269273\pi$
$578$	−12.5655	−0.522657
$579$	−30.2453	−1.25695
$580$	−1.49853	−0.0622229
$581$	−14.6425	−0.607475
$582$	11.1311	0.461397
$583$	0	0
$584$	13.6576	0.565156
$585$	3.95691	0.163598
$586$	−27.2901	−1.12735
$587$	2.43447	0.100481	0.0502407	−	0.998737i	$-0.484001\pi$
$587$	2.43447	0.100481	0.0502407	+	0.998737i	$0.484001\pi$
$588$	−9.67562	−0.399016
$589$	−3.06406	−0.126252
$590$	4.86894	0.200451
$591$	−20.1700	−0.829685
$592$	−4.65465	−0.191305
$593$	18.1841	0.746733	0.373367	−	0.927684i	$-0.378203\pi$
$593$	18.1841	0.746733	0.373367	+	0.927684i	$0.378203\pi$
$594$	0	0
$595$	3.65760	0.149947
$596$	−11.3453	−0.464723
$597$	−5.00295	−0.204757
$598$	−12.7828	−0.522726
$599$	4.33534	0.177137	0.0885687	−	0.996070i	$-0.471771\pi$
$599$	4.33534	0.177137	0.0885687	+	0.996070i	$0.471771\pi$
$600$	11.5906	0.473184
$601$	−16.2453	−0.662658	−0.331329	−	0.943515i	$-0.607497\pi$
$601$	−16.2453	−0.662658	−0.331329	+	0.943515i	$0.607497\pi$
$602$	−6.42235	−0.261755
$603$	−4.67267	−0.190286
$604$	14.6547	0.596289
$605$	0	0
$606$	−23.0950	−0.938171
$607$	−41.6995	−1.69253	−0.846266	−	0.532761i	$-0.821155\pi$
$607$	−41.6995	−1.69253	−0.846266	+	0.532761i	$0.821155\pi$
$608$	−1.00000	−0.0405554
$609$	15.7367	0.637684
$610$	−0.782765	−0.0316932
$611$	30.5685	1.23667
$612$	−14.7828	−0.597558
$613$	−48.1900	−1.94638	−0.973189	−	0.230008i	$-0.926125\pi$
$613$	−48.1900	−1.94638	−0.973189	+	0.230008i	$0.926125\pi$
$614$	−9.22313	−0.372215
$615$	0.995897	0.0401584
$616$	0	0
$617$	15.7187	0.632811	0.316406	−	0.948624i	$-0.397524\pi$
$617$	15.7187	0.632811	0.316406	+	0.948624i	$0.397524\pi$
$618$	16.6296	0.668940
$619$	44.6606	1.79506	0.897530	−	0.440954i	$-0.145360\pi$
$619$	44.6606	1.79506	0.897530	+	0.440954i	$0.145360\pi$
$620$	1.19922	0.0481617
$621$	2.31226	0.0927877
$622$	−6.34829	−0.254543
$623$	30.4102	1.21836
$624$	−8.89286	−0.355999
$625$	22.7258	0.909030
$626$	−13.3943	−0.535345
$627$	0	0
$628$	−10.4835	−0.418335
$629$	25.3093	1.00915
$630$	1.82880	0.0728612
$631$	2.38433	0.0949187	0.0474593	−	0.998873i	$-0.484888\pi$
$631$	2.38433	0.0949187	0.0474593	+	0.998873i	$0.484888\pi$
$632$	13.0029	0.517230
$633$	14.5685	0.579045
$634$	−31.0029	−1.23128
$635$	−1.95807	−0.0777035
$636$	−2.91087	−0.115424
$637$	15.0460	0.596146
$638$	0	0
$639$	2.84387	0.112502
$640$	0.391382	0.0154707
$641$	−45.6936	−1.80479	−0.902395	−	0.430910i	$-0.858193\pi$
$641$	−45.6936	−1.80479	−0.902395	+	0.430910i	$0.858193\pi$
$642$	6.34829	0.250547
$643$	42.2621	1.66666	0.833328	−	0.552779i	$-0.186433\pi$
$643$	42.2621	1.66666	0.833328	+	0.552779i	$0.186433\pi$
$644$	−5.90793	−0.232805
$645$	−3.49737	−0.137709
$646$	5.43742	0.213932
$647$	−38.7166	−1.52211	−0.761053	−	0.648690i	$-0.775317\pi$
$647$	−38.7166	−1.52211	−0.761053	+	0.648690i	$0.775317\pi$
$648$	9.76475	0.383595
$649$	0	0
$650$	−18.0239	−0.706956
$651$	−12.5935	−0.493580
$652$	−3.47346	−0.136031
$653$	38.6915	1.51412	0.757058	−	0.653348i	$-0.226636\pi$
$653$	38.6915	1.51412	0.757058	+	0.653348i	$0.226636\pi$
$654$	14.3483	0.561063
$655$	−2.54456	−0.0994243
$656$	−1.06406	−0.0415444
$657$	−37.1311	−1.44862
$658$	14.1281	0.550771
$659$	−12.7467	−0.496542	−0.248271	−	0.968691i	$-0.579862\pi$
$659$	−12.7467	−0.496542	−0.248271	+	0.968691i	$0.579862\pi$
$660$	0	0
$661$	−21.1671	−0.823305	−0.411652	−	0.911341i	$-0.635048\pi$
$661$	−21.1671	−0.823305	−0.411652	+	0.911341i	$0.635048\pi$
$662$	16.2993	0.633491
$663$	48.3542	1.87792
$664$	8.51949	0.330620
$665$	−0.672673	−0.0260851
$666$	12.6547	0.490358
$667$	−13.1612	−0.509604
$668$	−23.0950	−0.893573
$669$	−45.1370	−1.74510
$670$	−0.672673	−0.0259876
$671$	0	0
$672$	−4.11009	−0.158550
$673$	−2.25115	−0.0867755	−0.0433878	−	0.999058i	$-0.513815\pi$
$673$	−2.25115	−0.0867755	−0.0433878	+	0.999058i	$0.513815\pi$
$674$	−3.19217	−0.122958
$675$	3.26032	0.125490
$676$	0.828802	0.0318770
$677$	9.57848	0.368131	0.184065	−	0.982914i	$-0.441074\pi$
$677$	9.57848	0.368131	0.184065	+	0.982914i	$0.441074\pi$
$678$	−18.0059	−0.691512
$679$	−8.00000	−0.307012
$680$	−2.12811	−0.0816093
$681$	−50.1399	−1.92137
$682$	0	0
$683$	−32.0118	−1.22490	−0.612449	−	0.790510i	$-0.709815\pi$
$683$	−32.0118	−1.22490	−0.612449	+	0.790510i	$0.709815\pi$
$684$	2.71871	0.103952
$685$	−6.15203	−0.235057
$686$	18.9849	0.724848
$687$	33.0269	1.26005
$688$	3.73673	0.142461
$689$	4.52654	0.172448
$690$	−3.21724	−0.122478
$691$	−25.9699	−0.987940	−0.493970	−	0.869479i	$-0.664455\pi$
$691$	−25.9699	−0.987940	−0.493970	+	0.869479i	$0.664455\pi$
$692$	18.1771	0.690990
$693$	0	0
$694$	24.6966	0.937470
$695$	−4.35926	−0.165356
$696$	−9.15613	−0.347062
$697$	5.78571	0.219150
$698$	−4.43447	−0.167847
$699$	−11.7438	−0.444191
$700$	−8.33028	−0.314855
$701$	21.0950	0.796748	0.398374	−	0.917223i	$-0.369575\pi$
$701$	21.0950	0.796748	0.398374	+	0.917223i	$0.369575\pi$
$702$	−2.50147	−0.0944121
$703$	−4.65465	−0.175554
$704$	0	0
$705$	7.69364	0.289759
$706$	−25.3152	−0.952750
$707$	16.5986	0.624255
$708$	29.7497	1.11806
$709$	29.1981	1.09656	0.548278	−	0.836296i	$-0.315284\pi$
$709$	29.1981	1.09656	0.548278	+	0.836296i	$0.315284\pi$
$710$	0.409400	0.0153645
$711$	−35.3512	−1.32578
$712$	−17.6936	−0.663097
$713$	10.5324	0.394443
$714$	22.3483	0.836364
$715$	0	0
$716$	11.2842	0.421712
$717$	33.5965	1.25468
$718$	−13.9749	−0.521540
$719$	−13.9079	−0.518678	−0.259339	−	0.965786i	$-0.583505\pi$
$719$	−13.9079	−0.518678	−0.259339	+	0.965786i	$0.583505\pi$
$720$	−1.06406	−0.0396550
$721$	−11.9518	−0.445110
$722$	−1.00000	−0.0372161
$723$	−54.3793	−2.02239
$724$	21.8778	0.813082
$725$	−18.5575	−0.689209
$726$	0	0
$727$	−42.0059	−1.55791	−0.778956	−	0.627078i	$-0.784251\pi$
$727$	−42.0059	−1.55791	−0.778956	+	0.627078i	$0.784251\pi$
$728$	6.39138	0.236880
$729$	−21.7217	−0.804506
$730$	−5.34535	−0.197840
$731$	−20.3182	−0.751494
$732$	−4.78276	−0.176776
$733$	−36.0780	−1.33257	−0.666285	−	0.745697i	$-0.732117\pi$
$733$	−36.0780	−1.33257	−0.666285	+	0.745697i	$0.732117\pi$
$734$	25.3453	0.935514
$735$	3.78687	0.139681
$736$	3.43742	0.126705
$737$	0	0
$738$	2.89286	0.106488
$739$	−49.9988	−1.83924	−0.919619	−	0.392812i	$-0.871502\pi$
$739$	−49.9988	−1.83924	−0.919619	+	0.392812i	$0.871502\pi$
$740$	1.82175	0.0669689
$741$	−8.89286	−0.326687
$742$	2.09207	0.0768025
$743$	−0.220184	−0.00807777	−0.00403889	−	0.999992i	$-0.501286\pi$
$743$	−0.220184	−0.00807777	−0.00403889	+	0.999992i	$0.501286\pi$
$744$	7.32733	0.268633
$745$	4.44037	0.162683
$746$	8.95396	0.327828
$747$	−23.1620	−0.847454
$748$	0	0
$749$	−4.56258	−0.166713
$750$	−9.21608	−0.336524
$751$	−23.1311	−0.844064	−0.422032	−	0.906581i	$-0.638683\pi$
$751$	−23.1311	−0.844064	−0.422032	+	0.906581i	$0.638683\pi$
$752$	−8.22018	−0.299759
$753$	−28.6966	−1.04576
$754$	14.2382	0.518525
$755$	−5.73557	−0.208739
$756$	−1.15613	−0.0420480
$757$	37.4304	1.36043	0.680215	−	0.733013i	$-0.261886\pi$
$757$	37.4304	1.36043	0.680215	+	0.733013i	$0.261886\pi$
$758$	25.8527	0.939013
$759$	0	0
$760$	0.391382	0.0141969
$761$	33.9499	1.23068	0.615341	−	0.788261i	$-0.289018\pi$
$761$	33.9499	1.23068	0.615341	+	0.788261i	$0.289018\pi$
$762$	−11.9640	−0.433409
$763$	−10.3123	−0.373329
$764$	−7.09502	−0.256689
$765$	5.78571	0.209183
$766$	20.6296	0.745377
$767$	−46.2621	−1.67043
$768$	2.39138	0.0862916
$769$	44.7025	1.61201	0.806006	−	0.591907i	$-0.201625\pi$
$769$	44.7025	1.61201	0.806006	+	0.591907i	$0.201625\pi$
$770$	0	0
$771$	−46.4964	−1.67453
$772$	−12.6476	−0.455197
$773$	−44.8748	−1.61404	−0.807018	−	0.590527i	$-0.798920\pi$
$773$	−44.8748	−1.61404	−0.807018	+	0.590527i	$0.798920\pi$
$774$	−10.1591	−0.365161
$775$	14.8509	0.533461
$776$	4.65465	0.167092
$777$	−19.1311	−0.686323
$778$	−15.6086	−0.559596
$779$	−1.06406	−0.0381237
$780$	3.48051	0.124622
$781$	0	0
$782$	−18.6907	−0.668378
$783$	−2.57553	−0.0920419
$784$	−4.04604	−0.144501
$785$	4.10304	0.146444
$786$	−15.5475	−0.554561
$787$	−11.8418	−0.422113	−0.211056	−	0.977474i	$-0.567690\pi$
$787$	−11.8418	−0.422113	−0.211056	+	0.977474i	$0.567690\pi$
$788$	−8.43447	−0.300466
$789$	−35.0280	−1.24703
$790$	−5.08913	−0.181063
$791$	12.9410	0.460129
$792$	0	0
$793$	7.43742	0.264111
$794$	34.2512	1.21553
$795$	1.13927	0.0404056
$796$	−2.09207	−0.0741516
$797$	0.128110	0.00453789	0.00226895	−	0.999997i	$-0.499278\pi$
$797$	0.128110	0.00453789	0.00226895	+	0.999997i	$0.499278\pi$
$798$	−4.11009	−0.145496
$799$	44.6966	1.58125
$800$	4.84682	0.171361
$801$	48.1039	1.69967
$802$	−4.09207	−0.144496
$803$	0	0
$804$	−4.11009	−0.144952
$805$	2.31226	0.0814964
$806$	−11.3943	−0.401348
$807$	19.9640	0.702765
$808$	−9.65760	−0.339753
$809$	−17.5354	−0.616512	−0.308256	−	0.951304i	$-0.599745\pi$
$809$	−17.5354	−0.616512	−0.308256	+	0.951304i	$0.599745\pi$
$810$	−3.82175	−0.134283
$811$	39.3512	1.38181	0.690905	−	0.722946i	$-0.257212\pi$
$811$	39.3512	1.38181	0.690905	+	0.722946i	$0.257212\pi$
$812$	6.58060	0.230934
$813$	25.1992	0.883775
$814$	0	0
$815$	1.35945	0.0476194
$816$	−13.0029	−0.455194
$817$	3.73673	0.130732
$818$	2.48346	0.0868320
$819$	−17.3763	−0.607178
$820$	0.416452	0.0145431
$821$	−14.0980	−0.492023	−0.246011	−	0.969267i	$-0.579120\pi$
$821$	−14.0980	−0.492023	−0.246011	+	0.969267i	$0.579120\pi$
$822$	−37.5894	−1.31108
$823$	−1.12516	−0.0392207	−0.0196103	−	0.999808i	$-0.506243\pi$
$823$	−1.12516	−0.0392207	−0.0196103	+	0.999808i	$0.506243\pi$
$824$	6.95396	0.242253
$825$	0	0
$826$	−21.3814	−0.743953
$827$	33.8217	1.17610	0.588049	−	0.808825i	$-0.299896\pi$
$827$	33.8217	1.17610	0.588049	+	0.808825i	$0.299896\pi$
$828$	−9.34535	−0.324773
$829$	−36.1900	−1.25693	−0.628466	−	0.777837i	$-0.716317\pi$
$829$	−36.1900	−1.25693	−0.628466	+	0.777837i	$0.716317\pi$
$830$	−3.33438	−0.115738
$831$	54.7166	1.89810
$832$	−3.71871	−0.128923
$833$	22.0000	0.762255
$834$	−26.6355	−0.922311
$835$	9.03899	0.312807
$836$	0	0
$837$	2.06111	0.0712423
$838$	−35.0950	−1.21234
$839$	−31.8347	−1.09906	−0.549528	−	0.835475i	$-0.685192\pi$
$839$	−31.8347	−1.09906	−0.549528	+	0.835475i	$0.685192\pi$
$840$	1.60862	0.0555026
$841$	−14.3403	−0.494492
$842$	12.3483	0.425550
$843$	10.5015	0.361690
$844$	6.09207	0.209698
$845$	−0.324378	−0.0111590
$846$	22.3483	0.768350
$847$	0	0
$848$	−1.21724	−0.0418000
$849$	57.5534	1.97523
$850$	−26.3542	−0.903941
$851$	16.0000	0.548473
$852$	2.50147	0.0856991
$853$	31.1009	1.06488	0.532438	−	0.846469i	$-0.321276\pi$
$853$	31.1009	1.06488	0.532438	+	0.846469i	$0.321276\pi$
$854$	3.43742	0.117626
$855$	−1.06406	−0.0363899
$856$	2.65465	0.0907342
$857$	35.4985	1.21261	0.606303	−	0.795234i	$-0.292652\pi$
$857$	35.4985	1.21261	0.606303	+	0.795234i	$0.292652\pi$
$858$	0	0
$859$	−5.63760	−0.192352	−0.0961762	−	0.995364i	$-0.530661\pi$
$859$	−5.63760	−0.192352	−0.0961762	+	0.995364i	$0.530661\pi$
$860$	−1.46249	−0.0498705
$861$	−4.37336	−0.149044
$862$	−6.94691	−0.236613
$863$	31.9949	1.08912	0.544560	−	0.838722i	$-0.316697\pi$
$863$	31.9949	1.08912	0.544560	+	0.838722i	$0.316697\pi$
$864$	0.672673	0.0228848
$865$	−7.11420	−0.241890
$866$	13.2533	0.450364
$867$	30.0490	1.02052
$868$	−5.26622	−0.178747
$869$	0	0
$870$	3.58355	0.121494
$871$	6.39138	0.216564
$872$	6.00000	0.203186
$873$	−12.6547	−0.428295
$874$	3.43742	0.116272
$875$	6.62369	0.223921
$876$	−32.6606	−1.10350
$877$	3.23019	0.109076	0.0545378	−	0.998512i	$-0.482631\pi$
$877$	3.23019	0.109076	0.0545378	+	0.998512i	$0.482631\pi$
$878$	15.3152	0.516863
$879$	65.2612	2.20120
$880$	0	0
$881$	−24.7217	−0.832894	−0.416447	−	0.909160i	$-0.636725\pi$
$881$	−24.7217	−0.832894	−0.416447	+	0.909160i	$0.636725\pi$
$882$	11.0000	0.370389
$883$	11.9138	0.400932	0.200466	−	0.979701i	$-0.435754\pi$
$883$	11.9138	0.400932	0.200466	+	0.979701i	$0.435754\pi$
$884$	20.2202	0.680078
$885$	−11.6435	−0.391392
$886$	20.8807	0.701502
$887$	12.1841	0.409104	0.204552	−	0.978856i	$-0.434426\pi$
$887$	12.1841	0.409104	0.204552	+	0.978856i	$0.434426\pi$
$888$	11.1311	0.373534
$889$	8.59862	0.288388
$890$	6.92498	0.232126
$891$	0	0
$892$	−18.8748	−0.631976
$893$	−8.22018	−0.275078
$894$	27.1311	0.907398
$895$	−4.41645	−0.147626
$896$	−1.71871	−0.0574181
$897$	30.5685	1.02065
$898$	28.7887	0.960690
$899$	−11.7317	−0.391273
$900$	−13.1771	−0.439237
$901$	6.61862	0.220498
$902$	0	0
$903$	15.3583	0.511092
$904$	−7.52949	−0.250427
$905$	−8.56258	−0.284630
$906$	−35.0449	−1.16429
$907$	45.3211	1.50486	0.752431	−	0.658671i	$-0.228881\pi$
$907$	45.3211	1.50486	0.752431	+	0.658671i	$0.228881\pi$
$908$	−20.9669	−0.695811
$909$	26.2562	0.870864
$910$	−2.50147	−0.0829231
$911$	−13.6376	−0.451834	−0.225917	−	0.974147i	$-0.572538\pi$
$911$	−13.6376	−0.451834	−0.225917	+	0.974147i	$0.572538\pi$
$912$	2.39138	0.0791866
$913$	0	0
$914$	−32.7526	−1.08336
$915$	1.87189	0.0618828
$916$	13.8108	0.456321
$917$	11.1741	0.369003
$918$	−3.65760	−0.120719
$919$	−39.5283	−1.30392	−0.651960	−	0.758254i	$-0.726053\pi$
$919$	−39.5283	−1.30392	−0.651960	+	0.758254i	$0.726053\pi$
$920$	−1.34535	−0.0443547
$921$	22.0560	0.726771
$922$	20.2261	0.666111
$923$	−3.88991	−0.128038
$924$	0	0
$925$	22.5603	0.741777
$926$	18.6907	0.614214
$927$	−18.9058	−0.620948
$928$	−3.82880	−0.125687
$929$	6.89991	0.226379	0.113189	−	0.993573i	$-0.463893\pi$
$929$	6.89991	0.226379	0.113189	+	0.993573i	$0.463893\pi$
$930$	−2.86779	−0.0940384
$931$	−4.04604	−0.132604
$932$	−4.91087	−0.160861
$933$	15.1812	0.497010
$934$	−34.5685	−1.13112
$935$	0	0
$936$	10.1101	0.330459
$937$	21.3873	0.698692	0.349346	−	0.936994i	$-0.386404\pi$
$937$	21.3873	0.698692	0.349346	+	0.936994i	$0.386404\pi$
$938$	2.95396	0.0964503
$939$	32.0310	1.04529
$940$	3.21724	0.104935
$941$	27.5655	0.898611	0.449305	−	0.893378i	$-0.351672\pi$
$941$	27.5655	0.898611	0.449305	+	0.893378i	$0.351672\pi$
$942$	25.0700	0.816823
$943$	3.65760	0.119108
$944$	12.4404	0.404899
$945$	0.452489	0.0147195
$946$	0	0
$947$	−9.22313	−0.299712	−0.149856	−	0.988708i	$-0.547881\pi$
$947$	−9.22313	−0.299712	−0.149856	+	0.988708i	$0.547881\pi$
$948$	−31.0950	−1.00992
$949$	50.7887	1.64867
$950$	4.84682	0.157252
$951$	74.1399	2.40415
$952$	9.34535	0.302884
$953$	−22.6966	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$953$	−22.6966	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$954$	3.30931	0.107143
$955$	2.77687	0.0898573
$956$	14.0490	0.454377
$957$	0	0
$958$	−17.7187	−0.572466
$959$	27.0159	0.872389
$960$	−0.935945	−0.0302075
$961$	−21.6116	−0.697147
$962$	−17.3093	−0.558075
$963$	−7.21724	−0.232572
$964$	−22.7397	−0.732396
$965$	4.95005	0.159348
$966$	14.1281	0.454564
$967$	48.6966	1.56598	0.782988	−	0.622036i	$-0.213694\pi$
$967$	48.6966	1.56598	0.782988	+	0.622036i	$0.213694\pi$
$968$	0	0
$969$	−13.0029	−0.417715
$970$	−1.82175	−0.0584929
$971$	−19.5165	−0.626316	−0.313158	−	0.949701i	$-0.601387\pi$
$971$	−19.5165	−0.626316	−0.313158	+	0.949701i	$0.601387\pi$
$972$	−21.3332	−0.684264
$973$	19.1432	0.613702
$974$	−37.7426	−1.20935
$975$	43.1021	1.38037
$976$	−2.00000	−0.0640184
$977$	37.3512	1.19497	0.597486	−	0.801879i	$-0.296166\pi$
$977$	37.3512	1.19497	0.597486	+	0.801879i	$0.296166\pi$
$978$	8.30636	0.265608
$979$	0	0
$980$	1.58355	0.0505846
$981$	−16.3123	−0.520810
$982$	0.373364	0.0119145
$983$	−45.8888	−1.46362	−0.731812	−	0.681507i	$-0.761325\pi$
$983$	−45.8888	−1.46362	−0.731812	+	0.681507i	$0.761325\pi$
$984$	2.54456	0.0811177
$985$	3.30110	0.105182
$986$	20.8188	0.663006
$987$	−33.7857	−1.07541
$988$	−3.71871	−0.118308
$989$	−12.8447	−0.408438
$990$	0	0
$991$	−30.9418	−0.982900	−0.491450	−	0.870906i	$-0.663533\pi$
$991$	−30.9418	−0.982900	−0.491450	+	0.870906i	$0.663533\pi$
$992$	3.06406	0.0972838
$993$	−38.9779	−1.23693
$994$	−1.79783	−0.0570238
$995$	0.818801	0.0259577
$996$	−20.3734	−0.645555
$997$	−49.8418	−1.57850	−0.789252	−	0.614069i	$-0.789531\pi$
$997$	−49.8418	−1.57850	−0.789252	+	0.614069i	$0.789531\pi$
$998$	14.8388	0.469714
$999$	3.13106	0.0990623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4598.2.a.bm.1.3		3
11.10	odd	2		418.2.a.h.1.3	✓	3
33.32	even	2		3762.2.a.bd.1.3		3
44.43	even	2		3344.2.a.p.1.1		3
209.208	even	2		7942.2.a.bc.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.h.1.3	✓	3	11.10	odd	2
3344.2.a.p.1.1		3	44.43	even	2
3762.2.a.bd.1.3		3	33.32	even	2
4598.2.a.bm.1.3		3	1.1	even	1	trivial
7942.2.a.bc.1.1		3	209.208	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4598 = 2 \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4598.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.39138 q^{3} +1.00000 q^{4} -0.391382 q^{5} -2.39138 q^{6} +1.71871 q^{7} -1.00000 q^{8} +2.71871 q^{9} +0.391382 q^{10} +2.39138 q^{12} -3.71871 q^{13} -1.71871 q^{14} -0.935945 q^{15} +1.00000 q^{16} -5.43742 q^{17} -2.71871 q^{18} +1.00000 q^{19} -0.391382 q^{20} +4.11009 q^{21} -3.43742 q^{23} -2.39138 q^{24} -4.84682 q^{25} +3.71871 q^{26} -0.672673 q^{27} +1.71871 q^{28} +3.82880 q^{29} +0.935945 q^{30} -3.06406 q^{31} -1.00000 q^{32} +5.43742 q^{34} -0.672673 q^{35} +2.71871 q^{36} -4.65465 q^{37} -1.00000 q^{38} -8.89286 q^{39} +0.391382 q^{40} -1.06406 q^{41} -4.11009 q^{42} +3.73673 q^{43} -1.06406 q^{45} +3.43742 q^{46} -8.22018 q^{47} +2.39138 q^{48} -4.04604 q^{49} +4.84682 q^{50} -13.0029 q^{51} -3.71871 q^{52} -1.21724 q^{53} +0.672673 q^{54} -1.71871 q^{56} +2.39138 q^{57} -3.82880 q^{58} +12.4404 q^{59} -0.935945 q^{60} -2.00000 q^{61} +3.06406 q^{62} +4.67267 q^{63} +1.00000 q^{64} +1.45544 q^{65} -1.71871 q^{67} -5.43742 q^{68} -8.22018 q^{69} +0.672673 q^{70} +1.04604 q^{71} -2.71871 q^{72} -13.6576 q^{73} +4.65465 q^{74} -11.5906 q^{75} +1.00000 q^{76} +8.89286 q^{78} -13.0029 q^{79} -0.391382 q^{80} -9.76475 q^{81} +1.06406 q^{82} -8.51949 q^{83} +4.11009 q^{84} +2.12811 q^{85} -3.73673 q^{86} +9.15613 q^{87} +17.6936 q^{89} +1.06406 q^{90} -6.39138 q^{91} -3.43742 q^{92} -7.32733 q^{93} +8.22018 q^{94} -0.391382 q^{95} -2.39138 q^{96} -4.65465 q^{97} +4.04604 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{3} + 3 q^{4} + 5 q^{5} - q^{6} - q^{7} - 3 q^{8} + 2 q^{9} - 5 q^{10} + q^{12} - 5 q^{13} + q^{14} - 9 q^{15} + 3 q^{16} - 4 q^{17} - 2 q^{18} + 3 q^{19} + 5 q^{20} + 2 q^{23} - q^{24}+ \cdots + 6 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + q^3 + 3 * q^4 + 5 * q^5 - q^6 - q^7 - 3 * q^8 + 2 * q^9 - 5 * q^10 + q^12 - 5 * q^13 + q^14 - 9 * q^15 + 3 * q^16 - 4 * q^17 - 2 * q^18 + 3 * q^19 + 5 * q^20 + 2 * q^23 - q^24 + 4 * q^25 + 5 * q^26 - 2 * q^27 - q^28 - 7 * q^29 + 9 * q^30 - 3 * q^31 - 3 * q^32 + 4 * q^34 - 2 * q^35 + 2 * q^36 - 14 * q^37 - 3 * q^38 - 2 * q^39 - 5 * q^40 + 3 * q^41 + 5 * q^43 + 3 * q^45 - 2 * q^46 + q^48 - 6 * q^49 - 4 * q^50 - 2 * q^51 - 5 * q^52 - 16 * q^53 + 2 * q^54 + q^56 + q^57 + 7 * q^58 - 12 * q^59 - 9 * q^60 - 6 * q^61 + 3 * q^62 + 14 * q^63 + 3 * q^64 - 8 * q^65 + q^67 - 4 * q^68 + 2 * q^70 - 3 * q^71 - 2 * q^72 - 4 * q^73 + 14 * q^74 - 41 * q^75 + 3 * q^76 + 2 * q^78 - 2 * q^79 + 5 * q^80 - 17 * q^81 - 3 * q^82 - 7 * q^83 - 6 * q^85 - 5 * q^86 + 9 * q^87 + 16 * q^89 - 3 * q^90 - 13 * q^91 + 2 * q^92 - 22 * q^93 + 5 * q^95 - q^96 - 14 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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